Match each rule below with its corresponding graph. Can you do this without making any tables? Explain your selections. $\newline$ a. $y=-x^{2}-2$ $\newline$ b. $y=x^{2}-2$ $\newline$ c. $y=-x^{2}+2$ $\newline$ $1$ . $\newline$ $2$ . $\newline$ $3$ .

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Find the roots of factored polynomials

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Q. Match each rule below with its corresponding graph. Can you do this without making any tables? Explain your selections.

\newline

y=-x^{2}-2

\newline

y=x^{2}-2

\newline

y=-x^{2}+2

\newline

1

\newline

2

\newline

3

Understand Graph Shapes: The first step is to understand the general shape of the graphs of the given equations. All three equations are in the form of a quadratic function, $y = ax^2 + bx + c$ . The sign of the coefficient ' $a$ ' determines whether the parabola opens upwards ( $a > 0$ ) or downwards ( $a < 0$ ). The constant term ' $c$ ' determines the $y$ -intercept of the graph.
Analyze Equation a: Let's analyze equation a, $y = -x^2 - 2$ . Since the coefficient of $x^2$ is negative, the parabola opens downwards. The y-intercept is at $-2$ . This means the graph will be a downward-opening parabola that crosses the y-axis at $-2$ .
Analyze Equation b: Now let's look at equation b, $y = x^2 - 2$ . The coefficient of $x^2$ is positive, so the parabola opens upwards. The $y$ -intercept is also at $-2$ . This means the graph will be an upward-opening parabola that crosses the $y$ -axis at $-2$ .
Analyze Equation c: Finally, let's consider equation c, $y = -x^2 + 2$ . The coefficient of $x^2$ is negative, so the parabola opens downwards. The $y$ -intercept is at $2$ . This means the graph will be a downward-opening parabola that crosses the $y$ -axis at $2$ .
Match Rules to Graphs: Without seeing the actual graphs, we can match the rules to the graphs based on the direction they open (upward or downward) and their $y$ -intercepts. Graph $1$ should be the upward-opening parabola with a $y$ -intercept at $-2$ , which corresponds to equation $b$ . Graph $2$ should be the downward-opening parabola with a $y$ -intercept at $-2$ , which corresponds to equation $a$ . Graph $3$ should be the downward-opening parabola with a $y$ -intercept at $2$ , which corresponds to equation $1$ $2$ .